Transmitter diversity technique for wireless communications

ABSTRACT

A simple block coding arrangement is created with symbols transmitted over a plurality of transmit channels, in connection with coding that comprises only simple arithmetic operations, such as negation and conjugation. The diversity created by the transmitter utilizes space diversity and either time or frequency diversity. Space diversity is effected by redundantly transmitting over a plurality of antennas, time diversity is effected by redundantly transmitting at different times, and frequency diversity is effected by redundantly transmitting at different frequencies: Illustratively, using two transmit antennas and a single receive antenna, one of the disclosed embodiments provides the same diversity gain as the maximal-ratio receiver combining (MRRC) scheme with one transmit antenna and two receive antennas. The principles of this invention are applicable to arrangements with more than two antennas, and an illustrative embodiment is disclosed using the same space block code with two transmit and two receive antennas.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No. 13/740,827, filed Jan. 14, 2013, (now U.S. Pat. No. 8,767,874), which is a division of U.S. application Ser. No. 13/073,369, filed Mar. 28, 2011 (now U.S. Pat. No. 8,355,475), which is a continuation of U.S. application Ser. No. 11/828,790, filed Jul. 26, 2007 (now U.S. Pat. No. 7,916,806), which is a division of U.S. application Ser. No. 11/536,474, filed Sep. 28, 2006 (now U.S. Pat. No. 7,386,077), which is a continuation of U.S. application Ser. No. 10/873,567, filed Jun. 22, 2004 (now U.S. Pat. No. 7,120,200), which is a continuation of U.S. application Ser. No. 09/730,151, filed Dec. 5, 2000 (now U.S. Pat. No. 6,775,329), which is a continuation of U.S. application Ser. No. 09/074,224, filed May 7, 1998 (now U.S. Pat. No. 6,185,258), which claims the benefit of U.S. Provisional Application No. 60/059,016, filed Sep. 16, 1997, U.S. Provisional Application No. 60/059,219, filed Sep. 18, 1997,and U.S. Provisional Application No. 60/063,780, filed Oct. 31, 1997; all of which applications are hereby incorporated by reference herein in their entirety.

BACKGROUND OF THE INVENTION

This invention relates to wireless communication and, more particularly, to techniques for effective wireless communication in the presence of fading and other degradations.

The most effective technique for mitigating multipath fading in a wireless radio channel is to cancel the effect of fading at the transmitter by controlling the transmitter's power. That is, if the channel conditions are known at the transmitter (on one side of the link), then the transmitter can pre-distort the signal to overcome the effect of the channel at the receiver (on the other side). However, there are two fundamental problems with this approach. The first problem is the transmitter's dynamic range. For the transmitter to overcome an x dB fade, it must increase its power by x dB which, in most cases, is not practical because of radiation power limitations, and the size and cost of amplifiers. The second problem is that the transmitter does not have any knowledge of the channel as seen by the receiver (except for time division duplex systems, where the transmitter receives power from a known other transmitter over the same channel). Therefore, if one wants to control a transmitter based on channel characteristics, channel information has to be sent from the receiver to the transmitter, which results in throughput degradation and added complexity to both the transmitter and the receiver.

Other effective techniques are time and frequency diversity. Using time interleaving together with coding can provide diversity improvement. The same holds for frequency hopping and spread spectrum. However, time interleaving results in unnecessarily large delays when the channel is slowly varying. Equivalently, frequency diversity techniques are ineffective when the coherence bandwidth of the channel is large (small delay spread).

It is well known that in most scattering environments antenna diversity is the most practical and effective technique for reducing the effect of multipath fading. The classical approach to antenna diversity is to use multiple antennas at the receiver and perform combining (or selection) to improve the quality of the received signal.

The major problem with using the receiver diversity approach in current wireless communication systems, such as IS-136 and GSM, is the cost, size and power consumption constraints of the receivers. For obvious reasons, small size, weight and cost are paramount. The addition of multiple antennas and RF chains (or selection and switching circuits) in receivers is presently not feasible. As a result, diversity techniques have often been applied only to improve the up-link (receiver to base) transmission quality with multiple antennas (and receivers) at the base station. Since a base station often serves thousands of receivers, it is more economical to add equipment to base stations rather than the receivers.

Recently, some interesting approaches for transmitter diversity have been suggested. A delay diversity scheme was proposed by A. Wittneben in “Base Station Modulation Diversity for Digital SIMULCAST,” Proceedings of the 1991 IEEE Vehicular Technology Conference (VTC 41 st), pp. 848-853, May 1991, and in “A New Bandwidth Efficient Transmit Antenna Modulation Diversity Scheme For Linear Digital Modulation,” in Proceedings of the 1993 IEEE International Conference on Communications (IICC '93), pp. 1630-1634, May 1993. The proposal is for a base station to transmit a sequence of symbols through one antenna, and the same sequence of symbols—but delayed—through another antenna.

U.S. Pat. No. 5,479,448, issued to Nambirajan Seshadri on Dec. 26, 1995, discloses a similar arrangement where a sequence of codes is transmitted through two antennas. The sequence of codes is routed through a cycling switch that directs each code to the various antennas, in succession. Since copies of the same symbol are transmitted through multiple antennas at different times, both space and time diversity are achieved. A maximum likelihood sequence estimator (MLSE) or a minimum mean squared error (MMSE) equalizer is then used to resolve multipath distortion and provide diversity gain. See also N. Seshadri, J. H. Winters, “Two Signaling Schemes for Improving the Error Performance of FDD Transmission Systems Using Transmitter Antenna Diversity,” Proceedings of the 1993 IEEE Vehicular Technology Conference (VTC 43rd), pp. 508-511, May 1993; and J. H. Winters, “The Diversity Gain of Transmit Diversity in Wireless Systems with Rayleigh Fading,” Proceedings of the 1994 ICCSUPERCOMM, New Orleans, Vol. 2, pp. 1121-1125, May 1994.

Still another interesting approach is disclosed by Tarokh, Seshadri, Calderbank and Naguib in U.S. application Ser. No. 08/847635, filed Apr. 25, 1997 (based on a provisional application filed Nov. 7, 1996), where symbols are encoded according to the antennas through which they are simultaneously transmitted, and are decoded using a maximum likelihood decoder. More specifically, the process at the transmitter handles the information in blocks of M1 bits, where M1 is a multiple of M2, i.e., M1=k* M2. It converts each successive group of M2 bits into information symbols (generating thereby k information symbols), encodes each sequence of k information symbols into n channel codes (developing thereby a group of n channel codes for each sequence of k information symbols), and applies each code of a group of codes to a different antenna.

SUMMARY

The problems of prior art systems are overcome, and an advance in the art is realized with a simple block coding arrangement where symbols are transmitted over a plurality of transmit channels and the coding comprises only simple arithmetic operations, such as negation and conjugation. The diversity created by the transmitter utilizes space diversity and either time diversity or frequency diversity. Space diversity is effected by redundantly transmitting over a plurality of antennas; time diversity is effected by redundantly transmitting at different times; and frequency diversity is effected by redundantly transmitting at different frequencies. Illustratively, using two transmit antennas and a single receive antenna, one of the disclosed embodiments provides the same diversity gain as the maximal-ratio receiver combining (MRRC) scheme with one transmit antenna and two receive antennas. The novel approach does not require any bandwidth expansion or feedback from the receiver to the transmitter, and has the same decoding complexity as the MRRC. The diversity improvement is equal to applying maximal-ratio receiver combining (MRRC) at the receiver with the same number of antennas. The principles of this invention are applicable to arrangements with more than two antennas, and an illustrative embodiment is disclosed using the same space block code with two transmit and two receive antennas. This scheme provides the same diversity gain as four-branch MRRC.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a first embodiment in accordance with the principles of this invention;

FIG. 2 presents a block diagram of a second embodiment, where channel estimates are not employed;

FIG. 3 shows a block diagram of a third embodiment, where channel estimates are derived from recovered signals; and

FIG. 4 illustrates an embodiment where two transmitter antennas and two receiver antennas are employed.

DETAILED DESCRIPTION

In accordance with the principles of this invention, effective communication is achieved with encoding of symbols that comprises merely negations and conjugations of symbols (which really is merely negation of the imaginary part) in combination with a transmitter created diversity. Space diversity and either frequency diversity or time diversity are employed.

FIG. 1 presents a block diagram of an arrangement where the two controllable aspects of the transmitter that are used are space and time. That is, the FIG. 1 arrangement includes multiple transmitter antennas (providing space diversity) and employs multiple time intervals. Specifically, transmitter 10 illustratively comprises antennas 11 and 12, and it handles incoming data in blocks of n symbols, where n is the number of transmitter antennas, and in the illustrative embodiment of FIG. 1, it equals 2, and each block takes n symbol intervals to transmit. Also illustratively, the FIG. 1 arrangement includes a receiver 20 that comprises a single antenna 21.

At any given time, a signal sent by a transmitter antenna experiences interference effects of the traversed channel, which consists of the transmit chain, the air-link, and the receive chain. The channel may be modeled by a complex multiplicative distortion factor composed of a magnitude response and a phase response. In the exposition that follows therefore, the channel transfer function from transmit antenna 11 to receive antenna 21 is denoted by h_(o) and from transmit antenna 12 to receive antenna 21 is denoted by h₁ where: h₀=α₀e^(jθ) ⁰ h₁=α₁e^(jθ) ¹   (1) Noise from interference and other sources is added at the two received signals and, therefore, the resulting baseband signal received at any time and outputted by reception and amplification section 25 is r(t)=α₀ e ^(jθ) ⁰ s _(i)+α₁ e ^(jθ) ¹ s _(j) +n(t),   (2) where s_(i) and s_(j) are the signals being sent by transmit antenna 11 and 12, respectively.

As indicated above, in the two-antenna embodiment of FIG. 1 each block comprises two symbols and it takes two symbol intervals to transmit those two symbols. More specifically, when symbols s_(i) and s_(j) need to be transmitted, at a first time interval the transmitter applies signal s_(i) to antenna 11 and signal s_(j) to antenna 12, and at the next time interval the transmitter applies signal −s_(j)* to antenna 11 and signal s_(i)* to antenna 12. This is clearly a very simple encoding process where only negations and conjugations are employed. As demonstrated below, it is as effective as it is simple. Corresponding to the above-described transmissions, in the first time interval the received signal is r(t)=h ₀ s _(i) +h _(i) s _(j) +n(t),   (3) and in the next time interval the received signal is r(t+T)=h ₀ s _(j) *+h ₀ +n(t+T).   (4)

Table 1 illustrates the transmission pattern over the two antennas of the FIG. 1 arrangement for a sequence of signals {s₀, s₁, s₂, s₃, s₄, s₅ . . . }.

TABLE 1 Time: t t + T t + 2T t + 3T t + 4T t + 5T Antenna 11 s₀ −s₁* s₂ −s₃* s₄ −s₅* . . . Antenna 12 s₁  s₀* s₃  s₂* s₅  s₄* . . .

The received signal is applied to channel estimator 22, which provides signals representing the channel characteristics or, rather, the best estimates thereof. Those signals are applied to combiner 23 and to maximum likelihood detector 24. The estimates developed by channel estimator 22 can be obtained by sending a known training signal that channel estimator 22 recovers, and based on the recovered signal the channel estimates are computed. This is a well known approach.

Combiner 23 receives the signal in the first time interval, buffers it, receives the signal in the next time interval, and combines the two received signals to develop signals {tilde over (s)} _(i) ={tilde over (h)} ₀ *r(t)+{tilde over (h)} ₁ r*(t+T) {tilde over (s)} _(j) ={tilde over (h)} ₁ *r(t)−{tilde over (h)} ₀ r*(t+T)   (5) Substituting equation (1) into (5) yields {tilde over (s)} _(i)=({tilde over (α)} ₀ ²+{tilde over (α)} ₁ ²)s_(i) +{tilde over (h)} ₀ *n(t)+{tilde over (h)} ₁ n*(t+T) {tilde over (s)} _(j)=({tilde over (α)} ₀ ²+{tilde over (α)} ₁ ²)s_(j) +{tilde over (h)} ₀ n*(t+T)+{tilde over (h)} ₁ *n(t),   (6) where {tilde over (α)}₀ ²={tilde over (h)}₀{tilde over (h)}₀* and {tilde over (α)}₁ ²={tilde over (s)}₁{tilde over (s)}₁* demonstrating that the signals of equation (6) are, indeed, estimates of the transmitted signals (within a multiplicative factor). Accordingly, the signals of equation (6) are sent to maximum likelihood detector 24.

In attempting to recover s_(i), two kinds of signals are considered: the signals actually received at time t and t+T, and the signals that should have been received if s_(i) were the signal that was sent. As demonstrated below, no assumption is made regarding the value of That is, a decision is made that s_(i)=s_(x) for that value of x for which d ² [r(t), (h ₀ s _(x) +h ₁ s _(j))]+d ² [r(t+T), (−h ₁ s _(j) *+h ₀ s _(x)*)] is less than d ² [r(t), (h ₀ s _(k) +h ₁ s _(j))]+d ² [r(t+T), (−h ₁ s _(j) *+h ₀ s _(k)*)]  (7) where d² (x, y) is the squared Euclidean distance between signals x and y, i.e., d²(x, y)=|x−y|².

Recognizing that {tilde over (h)}₀={tilde over (h)}₀+noise that is independent of the transmitted symbol, and that {tilde over (h)}₁={tilde over (h)}₁+noise that is independent of the transmitted symbol, equation (7) can be rewritten to yield (α₀ ²+α₁ ²)|s _(x)|² −{tilde over (s)} _(i) s _(x) *−{tilde over (s)} _(i) *s _(x)≦(α₀ ²+α₁ ²)|s _(k)|² −{tilde over (s)} _(i) s _(k) *−{tilde over (s)} _(i) *s _(k)   (8) where α₀ ²=h₀h₀* and α₁ ²=h₁h₁*; or equivalently (α₀ ²+α₁ ²−1)|s _(x)|² +d ²({tilde over (s)} _(i) , s _(x))≦(α₀ ²+α₁ ²−1)|s _(k)|² +d ²({tilde over (s)} _(i) ,s _(k))   (9)

In Phase Shift Keying modulation, all symbols carry the same energy, which means that |s_(x)|²=|s_(k)|² and, therefore, the decision rule of equation (9) may be simplified to choose signal ŝ _(i) =s _(x)iff d ²({tilde over (s)} _(i) , s _(x))≦d ²({tilde over (s)} _(i) ,s _(k)).   (10) Thus, maximum likelihood detector 24 develops the signals s_(k) for all values of k, with the aid of {tilde over (h)}_(o) and {tilde over (h)}₁ from estimator 22, develops the distances d² ({tilde over (s)}_(i), s_(k)), identifies x for which equation (10) holds and concludes that ŝ_(i)=s_(x) . A similar process is applied for recovering ŝ_(j).

In the above-described embodiment each block of symbols is recovered as a block with the aid of channel estimates {tilde over (h)}₀ and {tilde over (h)}₁. However, other approaches to recovering the transmitted signals can also be employed. Indeed, an embodiment for recovering the transmitted symbols exists where the channel transfer functions need not be estimated at all, provided an initial pair of transmitted signals is known to the receiver (for example, when the initial pair of transmitted signals is prearranged). Such an embodiment is shown in FIG. 2, where maximum likelihood detector 27 is responsive solely to combiner 26. (Elements in FIG. 3 that are referenced by numbers that are the same as reference numbers in FIG. 1 are like elements.) Combiner 26 of receiver 30 develops the signals r ₀ =r(t)=h ₀ s ₀ +h ₁ s ₁ +n ₀ r ₁ =r(t+T)=h ₁ s ₀ *−h ₀ s ₁ *+n ₁ r ₂ =r(t+2T)=h ₀ s ₂ +h ₁ s ₃ +n ₂ r ₃ =r(t+3T)=h ₁ s ₂ *−h ₀ s ₃ *+n ₃,   (11) then develops intermediate signals A and B A=r ₀ r ₃ *−r ₂ r ₁* B=r ₂ r ₀ *−r ₁ r ₃*,   (12) and finally develops signals {tilde over (s)} ₂ =As ₁ *+Bs ₀ {tilde over (s)} ₃ =−As ₀ *+Bs ₁,   (13) where N₃ and N₄ are noise terms. It may be noted that signal r₂ is actually r₂=h₀ŝ₂+h₁ŝ₃=h₀s₂+h₁s₃+n₂, and similarly for signal r₃ . Since the makeup of signals A and B makes them also equal to A=(α₀ ²+α₁ ²)(s ₂ s ₁ −s ₃ s ₀)+N ₁ B=(α₀ ²+α₁ ²)(s ₂ s ₀ *+s ₃ s ₁*)+N ₂,   (14) where N1 and N2 are noise terms, it follows that signals {tilde over (s)}₂ and {tilde over (s)}₃ are equal to {tilde over (s)} ₂=(α₀ ²+α₁ ²)(|s ₀|² +|s ₁|²)s ₂ +N ₃ {tilde over (s)} ₃=(α₀ ²+α₁ ²)(|s ₀|² +|s ₁|²)s ₃ +N ₄.   (15)

When the power of all signals is constant (and normalized to 1) equation (15) reduces to {tilde over (s)} ₂=(α₀ ²+α₁ ²)s ₂ +N ₃ {tilde over (s)} ₃=(α₀ ²+α₁ ²)s ₃ +N ₄.   (16)

Hence, signals {tilde over (s)}₂ and {tilde over (s)}₃ are, indeed, estimates of the signals s₂ and s₃ (within a multiplicative factor). Lines 28 and 29 demonstrate the recursive aspect of equation (13), where signal estimates {tilde over (s)}₂ and {tilde over (s)}₃ are evaluated with the aid of recovered signals s₀ and s₁ that are fed back from the output of the maximum likelihood detector.

Signals {tilde over (s)}₂ and {tilde over (s)}₃ are applied to maximum likelihood detector where recovery is effected with the metric expressed by equation (10) above. As shown in FIG. 2, once signals s₂ and s₃ are recovered, they are used together with received signals r₂, r₃, r₄, and r₅ to recover signals s₄ and s₅, and the process repeats.

FIG. 3 depicts an embodiment that does not require the constellation of the transmitted signals to comprise symbols of equal power. (Elements in FIG. 3 that are referenced by numbers that are the same as reference numbers in FIG. 1 are like elements.) In FIG. 3, channel estimator 43 of receiver 40 is responsive to the output signals of maximum likelihood detector 42. Having access to the recovered signals s₀ and s₁, channel estimator 43 forms the estimates

$\begin{matrix} {{{\overset{\sim}{h}}_{0} = {\frac{{r_{0}s_{0}^{*}} - {r_{1}s_{1}}}{{s_{0}}^{2} + {s_{1}}^{2}} = {h_{0} + \frac{{s_{0}^{*}n_{0}} + {s_{1}n_{1}}}{{s_{0}}^{2} + {s_{1}}^{2}}}}}{{\overset{\sim}{h}}_{1} = {\frac{{r_{0}s_{1}^{*}} - {r_{1}s_{0}}}{{s_{0}}^{2} + {s_{1}}^{2}} = {h_{1} + \frac{{s_{1}^{*}n_{0}} + {s_{0}n_{1}}}{{s_{0}}^{2} + {s_{1}}^{2}}}}}} & (17) \end{matrix}$ and applies those estimates to combiner 23 and to detector 42. Detector 24 recovers signals s₂ and s₃ by employing the approach used by detector 24 of FIG. 1, except that it does not employ the simplification of equation (9). The recovered signals of detector 42 are fed back to channel estimator 43, which updates the channel estimates in preparation for the next cycle.

The FIGS. 1-3 embodiments illustrate the principles of this invention for arrangements having two transmit antennas and one receive antenna. However, those principles are broad enough to encompass a plurality of transmit antennas and a plurality of receive antennas. To illustrate, FIG. 4 presents an embodiment where two transmit antennas and two receive antennas are used; to wit, transmit antennas 31 and 32, and receive antennas 51 and 52. The signal received by antenna 51 is applied to channel estimator 53 and to combiner 55, and the signal received by antenna 52 is applied to channel estimator 54 and to combiner 55. Estimates of the channel transfer functions h₀ and h₁ are applied by channel estimator 53 to combiner 55 and to maximum likelihood detector 56. Similarly, estimates of the channel transfer functions h₂ and h₃ are applied by channel estimator 54 to combiner 55 and to maximum likelihood detector 56. Table 2 defines the channels between the transmit antennas and the receive antennas, and table 3 defines the notion for the received signals at the two receive antennas.

TABLE 2 Antenna 51 Antenna 52 Antenna 31 h₀ h₂ Antenna 32 h₁ h₃

TABLE 3 Antenna 51 Antenna 52 Time t r₀ r₂ Time t + T r₁ r₃

Based on the above, it can be shown that the received signals are r ₀ =h ₀ s ₀ +h ₁ s ₁ +n ₀ r ₁ =−h ₀ s ₁ *+h ₁ s ₀ *+n ₁ r ₂ =h ₂ s ₀ +h ₃ s ₁ +n ₂ r ₃ =−h ₂ s ₁ *+h ₃ s ₀ *+n ₃   (18) where n₀, n₁, n₂, and n₃ are complex random variables representing receiver thermal noise, interferences, etc.

In the FIG. 4 arrangement, combiner 55 develops the following two signals that are sent to the maximum likelihood detector: {tilde over (s)} ₀ =h ₀ *r ₀ +h ₁ r ₁ *+h ₂ *r ₂ +h ₃ r ₃* {tilde over (s)} ₁ =h ₁ *r ₀ −h ₀ r ₁ *+h ₃ *r ₂ −h ₂ r ₃*.   (19)

Substituting the appropriate equations results in {tilde over (s)} ₀=(α₀ ²+α₁ ²+α₂ ²+α₃ ²)s ₀ +h ₀ *n ₀ +h ₁ n ₁ *+h ₂ *n ₂ +h ₃* {tilde over (s)} ₁=(α₀ ²+α₁ ²+α₂ ²+α₃ ²)s ₁ +h ₁ *n ₀ −h ₀ n ₁ *+h ₃ *n ₂ −h ₂ n ₃*   (20) which demonstrates that the signal {tilde over (s)}₀ and {tilde over (s)}₁ are indeed estimates of the signals s₀ and s₁. Accordingly, signals {tilde over (s)}₀ and {tilde over (s)}₁ are sent to maximum likelihood decoder 56, which uses the decision rule of equation (10) to recover the signals ŝ₀ and ŝ₁.

As disclosed above, the principles of this invention rely on the transmitter to force a diversity in the signals received by a receiver, and that diversity can be effected in a number of ways. The illustrated embodiments rely on space diversity—effected through a multiplicity of transmitter antennas, and time diversity—effected through use of two time intervals for transmitting the encoded symbols. It should be realized that two different transmission frequencies could be used instead of two time intervals. Such an embodiment would double the transmission speed, but it would also increase the hardware in the receiver, because two different frequencies need to be received and processed simultaneously.

The above illustrated embodiments are, obviously, merely illustrative implementations of the principles of the invention, and various modifications and enhancements can be introduced by artisans without departing from the spirit and scope of this invention, which is embodied in the following claims. For example, all of the disclosed embodiments are illustrated for a space-time diversity choice, but as explained above, one could choose the space-frequency pair. Such a choice would have a direct effect on the construction of the receivers. 

The invention claimed is:
 1. A method of receiving a wireless communication from first and second space diverse antennas, the method comprising: estimating channel characteristics of communication channels between said first and second space diverse antennas and a receiver and supplying said estimated channel characteristics to a combiner and a detector; receiving at the receiver signals corresponding to a block of symbols that has been transmitted from said first and second space diverse antennas, the transmitted block of symbols including a first symbol, a second symbol, a complex conjugate of said first symbol, and a negative complex conjugate of the second symbol, wherein the first symbol and the complex conjugate of the first symbol are in respective signals received at different times and the second symbol and the negative complex conjugate of the second symbol are received in the respective signals at different times; supplying the received signals to the combiner and outputting estimates of transmitted symbols to the detector; and detecting in said detector the transmitted first and second symbols using said estimates and said estimated channel characteristics.
 2. A receiver comprising: a channel estimator to provide channel estimates between an antenna of the receiver and first and second space diverse transmitting antennas; a combiner responsive to signals received by said antenna and to said channel estimates developed for at least two transmission paths over which said signals arrive at said antenna from said first and second space diverse transmitting antennas, said combiner for developing symbol estimates by combining said signals received by said antenna with said channel estimates via operations that include multiplications and conjugations; and a detector coupled to receive said symbol estimates and to detect first and second transmitted symbols; wherein said signals received by said antenna represent a block of symbols including said first and second transmitted symbols and a complex conjugate of the first transmitted symbol and a negative complex conjugate of the second transmitted symbol; wherein the first symbol and the complex conjugate of the first symbol are received in respective signals received at different times and the second symbol and the negative complex conjugate of the second symbol are received in the respective signals received at different times. 